So, in the latest dev diary, we are told there are three "types of government", monarchy, oligarchy and democracy. In the above screenshot, which I have cropped, we are shown all the icons for the governments in game - with the "moral democracy" government type selected. This approach seems to me to derived from Classical Greek political thought, rather than the cluttered and poorly defined political ideologies of today. That is a good thing, I think, since it seems more timeless and universal, as befits a game set far in the future. Aliens won't have heard of Marx or Mussolini.
In the text of the diary, two specific types are mentioned, namely the military dictatorships and "Divine Mandate monarchies."
Your Ethos will limit your valid selection of government types, but there are always at least three to choose from; an oligarchy of some kind, a democracy or a monarchy. They all have their advantages and disadvantages. For example, in monarchies there are no elections, and you do not get to choose your successor when your ruler dies (except in Military Dictatorships), and if you die without an heir, all Factions in the empire will gain strength (oh, and there may be Pretender factions in monarchies...) On the other hand, each ruler may build a special "prestige object" in his or her lifetime, named after themselves. For example, military dictators can build a bigger, badder ship, and Divine Mandate monarchs can build a grand Mausoleum on a planet tile. Of course, both ethics and government types usually also have direct effects on the empire.
Lets break this down, and see what we can work out from the information we have been given.
First, lets look at how things are laid out. Seems to me, the top row of icons represent monarchies or more broadly regimes with a single ruler, note the lone figure of a military leader at a podium and a mediating figure who appears to be enlightened in some way, some kind of prophet or high priest. I think this is the "Divine Mandate" system of government. The symbolism of a crown in a lightbulb is very obvious, as is the significance of the Double-Headed bird, which in our earth cultures means only one thing, the Imperial legacy of Byzantium/Rome - indicating Empire. The second row done is Oligarchy, i.e. rule by a small and select group, as represented by what appears to be a group of generals next to a group of Catholic priests (note the signature neck collars) and a cartoonish Mr Moneybags capitalist at the other end. The third row down is Democracy, rule-by-many, firstly because we know it has "Moral Democracy" selected and because of the obvious symbolism of ballot papers.
Additionally, from left to right, it seems to progress from military type governments on the far left (all generals), to theocratic (all priests) to what appears to be some kind of technocratic or scientific category in the middle, as hinted at by atomic symbol. Next is what appears to be the "moral" category, where we have "Moral Democracy", which is the one government we know most about. I'm not really sure what is next. The eagle, as I said, looks very Imperial, the capitalist obviously very capitalistic, but what exactly the circular arrow on a ballot is supposed to be is hard to tell, though obviously it is some manner of democracy. Provisionally, I think this is some kind of "wealth" based category of governments, so I've named it capitalist after Mr Moneybags the Space Capitalist for now.

Heres a chart that I think sums it up. What do you guys think?
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